Pade Table Research Articles

On Overconvergence of Closed to Row Subsequences of Classical Padé Approximants

Let $$f(z) := \sum f_\nu z^\nu $$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade approximants $$\{\pi _{n,m_n}\}$$ associated with f, where $$m_n\rightarrow \infty $$, $$m_n\le m_{ n+1}\le m_n+1$$ and $$m_n = o(n/\log n)$$, resp. $$m_n = o(n)$$ as $$n\rightarrow \infty $$. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lopez Lagomasino and A. Fernandes Infante concerning overconvergent subsequences of a fixed row of the Pade table.

Multivariate homogeneous two-point Padé approximants and continued fractions

The multivariate homogeneous two-point Pade approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogonality conditions. Diverse three-term recurrence relations satisfied by the monic orthogonal polynomials are presented. Various continued fractions provided by these relations and the quotient-difference algorithm applied to a power series (positive or negative exponents) are described in terms of their relationships with the multivariate homogeneous two-point Pade table. Numerical examples are furnished to illustrate our results.

Remainder Pade Approximants for the Exponential Function

Following our earlier research, we propose a new method for obtaining the complete Pade table of the exponential function. It is based on an explicit construction of certain Pade approximants, not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and nontrivial coincidence is proved more generally for type II simultaneous Pade approximants for a family \((\exp(a_jz))_{j=1,\ldots, r}\) with distinct complex a's and we recover Hermite's classical formulas. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesu, Coussement, and van Assche, which generalize the classical Charlier orthogonal polynomials.

Explicit construction of general multivariate Pad� approximants to an Appell function

Properties of Pade approximants to the Gauss hypergeometric function 2F1(a,b;c;z) have been studied in several papers and some of these properties have been generalized to several variables in [6]. In this paper we derive explicit formulae for the general multivariate Pade approximants to the Appell function F1(a,1,1;a+1;x,y)=∑ i,j=0 ∞ (axiyj/(i+j+a)), where a is a positive integer. In particular, we prove that the denominator of the constructed approximant of partial degree n in x and y is given by \(q(x,y)=(-1)^{n}{{m+n+a}\choose n}F_{1}(-m-a,-n,-n;-m-n-a;x,y)\) , where the integer m, which defines the degree of the numerator, satisfies m≥n+1 and m+a≥2n. This formula generalizes the univariate explicit form for the Pade denominator of 2F1(a,1;c;z), which holds for c>a>0 and only in half of the Pade table. From the explicit formulae for the general multivariate Pade approximants, we can deduce the normality of a particular multivariate Pade table.

A New Method for Deriving Adjacent Approximants in the Padé Table

This paper derives formulas for adjacent approximants in the Pade table by using the Focal Point method of solving systems of linear equations. In addition to presenting a new method to derive formulas, this paper gives inner products to detect blocks in a nonnormal Pade table. These inner products lead to characterizations of the corner entries of the blocks.

A Rational Triplic Form for the Exponential

The theory of the power series solution is applied to the Hamburger DE to derive an improper fraction triplic form, E2N(x), for ex on the real line. E2N turns out to be a simpler equivalent to the vertical sequence P2N of the related Pade table PMN and with comparable summation features.

Generalized Moment Representations and Padé Approximants Associated with Bilinear Transformations

By using the method of generalized moment representations with an operator of bilinear transformation of an independent variable, we construct elements of the first subdiagonal of the Pade table for certain special power series.

Padé approximants for the numerical solution of singular integral equations

A method for accelerating the convergence of the numerical solution of a singular integral equation, based on Pade Approximants, is given in this paper. At first the general form of the Pade Table and of the “epsilon” algorithm are presented. Taking into consideration the classical quadrature method, based on the Gauss-Jacobi quadrature rule, an approximate formula is derived for the unknown density function of the Cauchy-type singular integral equation or of the equivalent Fredholm integral equation. In this formula applying the “epsilon” algorithm to the solution for the stress intensity factors, the convergence is achieved after a few operations. The number of numerical operations required for the determination of stress intensity factors is considerable reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method.

The Diagonal of the Padé Table and the Approximation of the Weyl Function of Second-Order Difference Operators

We study connections between diagonal Pade approximants and spectral properties of second-order difference operators with complex coefficients. In the first part, we identify the diagonal of the Pade table with a particular difference operator which is shown to have a maximal resolvent set. The spectrum of an asymptotically periodic complex difference operator is given, and we prove convergence on the resolvent set for the corresponding sequence of Pade approximants to the associated Weyl function.

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Pade approximants connects neighbouring elements called N, S, E, W and C in the Pade table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case.

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Pade Approximants of the given power series. Furthermore, given a matrix rational function, the “smallest” degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Pade Table.

A "Look-around Lanczos" algorithm for solving a system of linear equations

Two algorithms for the solution of a large sparse linear system of equations are proposed. The first is a modification of Lanczos' method and the second is based on one of Brezinski's methods. Both the latter methods are iterative and they can break down. In practical situations, serious numerical error is far more likely to occur because an ill-conditioned pair of polynomials is (implicitly) used in the calculation rather than complete breakdown arising because a large square block of exactly defective polynomials is encountered. The algorithms proposed use a method based on selecting well-conditioned pairs of neighbouring polynomials (in the associated Pade table), and the method is equivalent to going round the blocks instead of going across them, as is done in the well-known look-ahead methods.

Orthogonal polynomials, Padé approximations and A-stability

This paper consists of a brief survey of implicit Runge-Kutta methods based on Gauss-Legendre and closely related quadrature formulas, together with a new proof of a classical result. The classical result concerns the A-stability of those implicit Runge-Kutta methods whose stability functions are Pade approximations to the exponential function and lie on the diagonal and the first two subdiagonals of the Pade table.

Kronecker type theorems, normality and continuity of the multivariate Padé operator

For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Pade approximants and the approximated function \(f\) being rational, is well-known. In [Lubi88] Lubinsky proved that if \(f\) is not rational, then its Pade table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of \(f\) is such that it generates a non-normal Pade table. This implies that the Pade operator is an almost always continuous operator because it is continuous when computing a normal Pade approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Pade approximation. We distinguish between two different approaches for the definition of multivariate Pade approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84].

Pade table, continued fraction expansion, and perfect reconstruction filter banks

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence.

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Pade approximants occupying the diagonal of the Pade table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.

The multipoint Pad� table and general recurrences for rational interpolation

We first review briefly the Newton-Pade approximation problem and the analogous problem with additional interpolation conditions at infinity, which we call multipoint Pade approximation problem. General recurrence formulas for the Newton-Pade table combine either two pairs of Newton-Pade forms or one such pair and a pair of multipoint Pade forms. We show that, likewise, certain general recurrences for the multipoint Pade table compose two pairs of multipoint Pade forms to get a new pair of multipoint Pade forms. We also discuss the possibility of superfast, i.e.,O(n log2n) algorithms for certain rational interpolation problems.

Two Families of Approximation Schemes for Delay Systems

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Pade fractions in the main diagonal resp. in the diagonal below the main diagonal of the Pade table for the exponential function play an essential role.

On the methods for solving Yule-Walker equations

The three well-known fast algorithms for the solution of Yule-Walker equations-the Levinson, Euclidean, and Berlekamp-Massey algorithms-are reviewed, and the relationship between each of them and the Pade approximation problem is shown. The algorithms are classified with reference to three criteria, namely: (1) the path they follow in the Pade table; (2) the organization of the computations (one-pass or two-pass); and (3) the auxiliary variables used. This classification shows that the set of known classical algorithms is not complete, and the missing variants are proposed. With these variants of the Berlekamp-Massey and Euclid algorithms, both forward and backward predictors can be obtained without additional cost. A unified representation of the two-pass algorithms is given in such a way that the application of the divide and conquer strategy becomes straightforward. A general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The structure of the singular solution table of the M-Padé approximation problem

A general interpolation problem is defined which contains as special cases the M-Padé approximation problem as well as classical interpolation problems, e.g., Newton-Padé approximation. To study the structure of the solution table a new concept—the minimal solution—is introduced; statements on the local and global structure of the solution table are obtained. As a special case the well-known block structure of the Padé table is derived. Finally, a characterization of the solution set is given.